Comments Julien
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@ -192,7 +192,7 @@ Here, we only provide the main working equations.
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We refer the interested reader to Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} for a more formal derivation.
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Let us assume we have both the energy $\E{\modY}{\Bas}$ and density $\n{\modZ}{\Bas}$ of a $\Ne$-electron system described by two methods $\modY$ and $\modZ$ (potentially identical) in an incomplete basis set $\Bas$.
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According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\E{\modY}{\Bas}$ and $\n{\modZ}{\Bas}$ are reasonable approximations of the FCI energy and density within $\Bas$, the exact ground state energy $\E{}{}$ may be \titou{approximated} as
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According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\E{\modY}{\Bas}$ and $\n{\modZ}{\Bas}$ are reasonable approximations of the FCI energy and density within $\Bas$, the exact ground state energy $\E{}{}$ may be approximated as
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\begin{equation}
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\label{eq:e0basis}
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\E{}{}
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@ -208,7 +208,7 @@ where
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\end{equation}
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is the basis-dependent complementary density functional, $\hT$ is the kinetic operator and $\hWee{} = \sum_{i<j} r_{ij}^{-1}$ is the interelectronic repulsion operator.
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In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Ne$-electron normalized wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis set (CBS), respectively.
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Both wave functions yield the same target density $\n{}{}$ (assumed to be representable in \trashPFL{the basis set} $\Bas$).
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Both wave functions yield the same target density $\n{}{}$ (assumed to be representable in $\Bas$).
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Importantly, in the CBS limit (which we refer to as $\Bas \to \infty$), we have, for any density $\n{}{}$, $\lim_{\Bas \to \infty} \bE{}{\Bas}[\n{}{}] = 0$.
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This implies that
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\begin{equation}
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@ -217,18 +217,18 @@ This implies that
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\end{equation}
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where $\E{\modY}{}$ is the energy associated with the method $\modY$ in the CBS limit.
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In the case where $\modY = \FCI$ in Eq.~\eqref{eq:limitfunc}, we have a strict equality as $\E{\FCI}{} = \E{}{}$.
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Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only source of error at this stage lies in the potential approximate nature of the methods $\modY$ and $\modZ$, \titou{and the lack of self-consistency.}
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Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only source of error at this stage lies in the potential approximate nature of the methods $\modY$ and $\modZ$, and the lack of self-consistency.
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The functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$.
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Moreover, as $\bE{}{\Bas}[\n{}{}]$ aims at fixing the incompleteness of $\Bas$, its main role is to correct
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for the lack of \titou{cusp (i.e.~discontinuous derivative) in $\wf{}{\Bas}$} at the e-e coalescence points, a universal condition of exact wave functions.
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for the lack of cusp (i.e.~discontinuous derivative) in $\wf{}{\Bas}$ at the e-e coalescence points, a universal condition of exact wave functions.
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Because the e-e cusp originates from the divergence of the Coulomb operator at $r_{12} = 0$, a cuspless wave function could equivalently originate from a Hamiltonian with a non-divergent two-electron interaction at coalescence.
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Therefore, as we shall do later on, it feels natural to approximate $\bE{}{\Bas}[\n{}{}]$ by a short-range density functional which is complementary to a non-divergent long-range interaction.
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Contrary to the conventional RS-DFT scheme which requires a range-separation \textit{parameter} $\rsmu{}{}$, here we use a range-separation \textit{function} $\rsmu{}{\Bas}(\br{})$ that automatically adapts to quantify the incompleteness of $\Bas$ in $\mathbb{R}^3$.
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The first step of the present basis-set correction consists of obtaining an effective two-electron interaction $\W{}{\Bas}(\br{1},\br{2})$ ``mimicking'' the Coulomb operator in an incomplete basis $\Bas$.
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In a second step, we shall link $\W{}{\Bas}(\br{1},\br{2})$ to $\rsmu{}{\Bas}(\br{})$.
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\titou{As a} final step, we employ short-range density functionals \cite{TouGorSav-TCA-05} with $\rsmu{}{\Bas}(\br{})$ as range-separation function.
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As a final step, we employ short-range density functionals \cite{TouGorSav-TCA-05} with $\rsmu{}{\Bas}(\br{})$ as range-separation function.
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%=================================================================
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%\subsection{Effective Coulomb operator}
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@ -262,10 +262,10 @@ With such a definition, $\W{}{\Bas}(\br{1},\br{2})$ satisfies (see Appendix A of
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\mel*{\wf{}{\Bas}}{\hWee{\updw}}{\wf{}{\Bas}} = \frac{1}{2}\iint \W{}{\Bas}(\br{1},\br{2}) \n{2}{\Bas}(\br{1},\br{2}) \dbr{1} \dbr{2},
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\end{equation}
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where $\hWee{\updw}$ contains only the opposite-spin component of $\hWee{}$.
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Because Eq.~\eqref{eq:int_eq_wee} can be \titou{recast} as
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Because Eq.~\eqref{eq:int_eq_wee} can be recast as
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\begin{equation}
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\alert{\iint \frac{ \n{2}{\Bas}(\br{1},\br{2})}{r_{12}} \dbr{1} \dbr{2} =
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\iint \W{}{\Bas}(\br{1},\br{2}) \n{2}{\Bas}(\br{1},\br{2}) \dbr{1} \dbr{2},}
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\iint \frac{ \n{2}{\Bas}(\br{1},\br{2})}{r_{12}} \dbr{1} \dbr{2} =
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\iint \W{}{\Bas}(\br{1},\br{2}) \n{2}{\Bas}(\br{1},\br{2}) \dbr{1} \dbr{2},
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\end{equation}
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it intuitively motivates $\W{}{\Bas}(\br{1},\br{2})$ as a potential candidate for an effective interaction.
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Note that the divergence condition of $\W{}{\Bas}(\br{1},\br{2})$ in Eq.~\eqref{eq:def_weebasis} ensures that one-electron systems are free of correction as the present approach must only correct the basis set incompleteness error originating from the e-e cusp.
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@ -273,7 +273,7 @@ As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{}{\Bas}
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Thanks to its definition one can show that (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
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\begin{equation}
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\label{eq:lim_W}
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\lim_{\Bas \to \infty}\W{}{\Bas}(\br{1},\br{2}) = \titou{r_{12}^{-1} }
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\lim_{\Bas \to \infty}\W{}{\Bas}(\br{1},\br{2}) = r_{12}^{-1},
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\end{equation}
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for any $(\br{1},\br{2})$ such that $\n{2}{\Bas}(\br{1},\br{2}) \ne 0$.
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@ -289,11 +289,7 @@ Although this choice is not unique, we choose here the range-separation function
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\label{eq:mu_of_r}
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\rsmu{}{\Bas}(\br{}) = \frac{\sqrt{\pi}}{2} \W{}{\Bas}(\br{},\br{}),
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\end{equation}
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such that the long-range interaction of RS-DFT, \titou{$\w{}{\lr,\mu}(r_{12}) = \erf( \mu r_{12})/r_{12}$},
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%\begin{equation}
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% \w{}{\lr,\mu}(r_{12}) = \frac{\erf( \mu r_{12})}{r_{12}}
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%\end{equation}
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coincides with the effective interaction at coalescence, i.e.~$\w{}{\lr,\rsmu{}{\Bas}(\br{})}(0) = \W{}{\Bas}(\br{},\br{})$ at any \trashPFL{point} $\br{}$.
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such that the long-range interaction of RS-DFT, $\w{}{\lr,\mu}(r_{12}) = \erf( \mu r_{12})/r_{12}$, coincides with the effective interaction at coalescence, i.e.~$\w{}{\lr,\rsmu{}{\Bas}(\br{})}(0) = \W{}{\Bas}(\br{},\br{})$ at any $\br{}$.
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%=================================================================
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%\subsection{Short-range correlation functionals}
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@ -315,44 +311,45 @@ with $\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})$.
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The ECMD functionals admit, for any $\n{}{}$, the following two limiting forms
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\begin{align}
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\label{eq:large_mu_ecmd}
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\lim_{\mu \to \infty} \bE{\titou{\text{c,md}}}{\sr}[\n{}{},\rsmu{}{}] & = 0,
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\lim_{\mu \to \infty} \bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}] & = 0,
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&
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% \label{eq:small_mu_ecmd}
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\lim_{\mu \to 0} \bE{\titou{\text{c,md}}}{\sr}[\n{}{},\rsmu{}{}] & = \Ec[\n{}{}],
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\lim_{\mu \to 0} \bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}] & = \Ec[\n{}{}],
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\end{align}
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where $\Ec[\n{}{}]$ is the usual universal correlation density functional defined in KS-DFT.
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The choice of \trashPFL{the} ECMD in the present scheme is motivated by the analogy between the definition of $\bE{}{\Bas}[\n{}{}]$ [Eq.~\eqref{eq:E_funcbasis}] and the ECMD functional [Eq.~\eqref{eq:ec_md_mu}].
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The choice of ECMD in the present scheme is motivated by the analogy between the definition of $\bE{}{\Bas}[\n{}{}]$ [Eq.~\eqref{eq:E_funcbasis}] and the ECMD functional [Eq.~\eqref{eq:ec_md_mu}].
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Indeed, the two functionals coincide if $\wf{}{\Bas} = \wf{}{\rsmu{}{}}$.
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Therefore, we approximate $\bE{}{\Bas}[\n{}{}]$ by ECMD functionals evaluated with the range-separation function $\rsmu{}{\Bas}(\br{})$.
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The local-density approximation (LDA) of the ECMD complementary functional is defined as
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\begin{equation}
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\label{eq:def_lda_tot}
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\bE{\LDA}{\Bas}[\n{}{},\rsmu{}{\Bas}] = \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\LDA}\qty(\qty{\n{\sigma}{}(\br{})},\rsmu{}{\Bas}(\br{})) \dbr{},
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\titou{\bE{\LDA}{\Bas}[\n{}{},\zeta,\rsmu{}{\Bas}] = \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\LDA}\qty(\n{}{}(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},}
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\end{equation}
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where $\be{\text{c,md}}{\sr,\LDA}(\qty{\n{\sigma}{}},\rsmu{}{})$ is the ECMD correlation energy per electron of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parametrized in Ref.~\onlinecite{PazMorGorBac-PRB-06} as a function of the spin densities $\qty{\n{\sigma}{}}_{\sigma=\uparrow,\downarrow}$ and the range-separation parameter $\mu$.
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where \titou{$\zeta = (\n{\uparrow}{} - \n{\downarrow}{})/\n{}{}$ is the spin-polarization and $\be{\text{c,md}}{\sr,\LDA}(\n{}{},\zeta,\rsmu{}{})$} is the ECMD correlation energy per electron of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parametrized in Ref.~\onlinecite{PazMorGorBac-PRB-06} as a function of the spin densities $\qty{\n{\sigma}{}}_{\sigma=\uparrow,\downarrow}$ and the range-separation parameter $\mu$.
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The short-range LDA correlation functional relies on the transferability of the physics of the UEG which is certainly valid for large $\mu$ but is known to over correlate for small $\mu$.
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In order to correct such a defect, we propose here a new Perdew-Burke-Ernzerhof (PBE)-based ECMD functional
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\begin{multline}
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\label{eq:def_pbe_tot}
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\bE{\PBE}{\Bas}[\n{}{},\rsmu{}{\Bas}] =
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\titou{\bE{\PBE}{\Bas}[\n{}{},s,\zeta,\rsmu{}{\Bas}] =}
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\\
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\int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\PBE}\qty(\qty{\n{\sigma}{}(\br{})},\qty{\nabla \n{\sigma}{}(\br{})},\rsmu{}{\Bas}(\br{})) \dbr{},
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\titou{\int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},}
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\end{multline}
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inspired by the recent functional proposed by some of the authors \cite{FerGinTou-JCP-18} which interpolates between the usual PBE correlation functional, \cite{PerBurErn-PRL-96} $\e{\text{c}}{\PBE}(\qty{\n{\sigma}{}},\qty{\nabla \n{\sigma}{}})$, \titou{at} $\rsmu{}{}=0$ and the exact large-$\rsmu{}{}$ behavior, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGorBac-PRB-06} yielding
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\titou{(where $s$ is the reduced gradient)} inspired by the recent functional proposed by some of the authors. \cite{FerGinTou-JCP-18}
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\titou{$\be{\text{c,md}}{\sr,\PBE}\qty(\n{}{},s,\zeta,\rsmu{}{})$} interpolates between the usual PBE correlation functional, \cite{PerBurErn-PRL-96} \titou{$\e{\text{c}}{\PBE}(\n{}{},s,\zeta)$}, at $\rsmu{}{}=0$ and the exact large-$\rsmu{}{}$ behavior, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGorBac-PRB-06} yielding
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\begin{subequations}
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\begin{gather}
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\label{eq:epsilon_cmdpbe}
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\be{\text{c,md}}{\sr,\PBE}(\qty{\n{\sigma}{}},\qty{\nabla \n{\sigma}{}},\rsmu{}{}) = \frac{\e{\text{c}}{\PBE}(\qty{\n{\sigma}{}},\qty{\nabla \n{\sigma}{}})}{1 + \beta(\qty{n_\sigma},\qty{\nabla n_\sigma}) \rsmu{}{3} },
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\titou{\be{\text{c,md}}{\sr,\PBE}(\n{}{},s,\zeta,\rsmu{}{}) = \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{1 + \beta(\n{}{},s,\zeta) \rsmu{}{3} },}
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\\
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\label{eq:beta_cmdpbe}
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\beta(\qty{n_\sigma},\qty{\nabla n_\sigma}) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\text{c}}{\PBE}(\qty{\n{\sigma}{}},\qty{\nabla \n{\sigma}{}})}{\n{2}{\UEG}(0,\qty{\n{\sigma}{}})}.
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\titou{\beta(\n{}{},s,\zeta) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{\n{2}{\UEG}(0,\n{}{},\zeta)}.}
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\end{gather}
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\end{subequations}
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The difference between the ECMD functional defined in Ref.~\onlinecite{FerGinTou-JCP-18} and the present expression \eqref{eq:epsilon_cmdpbe}-\eqref{eq:beta_cmdpbe} is that we approximate here the on-top pair density by its UEG version, i.e.~$\n{2}{\Bas}(\br{},\br{}) \approx \n{2}{\UEG}(0,\qty{\n{\sigma}{}(\br{})})$, where $0$ refers to $r_{12}=0$ and $\n{2}{\UEG}(0,\qty{n_\sigma}) \approx 4 \; n_{\uparrow} \; n_{\downarrow} \; g(0,n)$ with the parametrization of the UEG on-top pair-distribution function $g(0,n)$ given in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}.
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The difference between the ECMD functional defined in Ref.~\onlinecite{FerGinTou-JCP-18} and the present expression \eqref{eq:epsilon_cmdpbe}-\eqref{eq:beta_cmdpbe} is that we approximate here the on-top pair density by its UEG version, i.e.~\titou{$\n{2}{\Bas}(\br{},\br{}) \approx \n{2}{\UEG}(0,\n{}{}(\br{}),\zeta(\br{}))$, where $\n{2}{\UEG}(r_{12},\n{}{},\zeta) \approx 4 \n{\uparrow}{} \n{\downarrow}{} g(r_{12},n)$} with the parametrization of the UEG on-top pair-distribution function $g(0,n)$ given in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}.
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This represents a major computational saving without loss of accuracy for weakly correlated systems as we eschew the computation of $\n{2}{\Bas}(\br{},\br{})$.
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Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\modZ}{\Bas}]$ is \titou{approximated by} $\bE{\LDA}{\Bas}[\n{\modZ}{\Bas},\rsmu{}{\Bas}]$ or $\bE{\PBE}{\Bas}[\n{\modZ}{\Bas},\rsmu{}{\Bas}]$ where $\rsmu{}{\Bas}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.
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Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\modZ}{\Bas}]$ is approximated by $\bE{\LDA}{\Bas}[\n{\modZ}{\Bas},\rsmu{}{\Bas}]$ or $\bE{\PBE}{\Bas}[\n{\modZ}{\Bas},\rsmu{}{\Bas}]$ where $\rsmu{}{\Bas}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.
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%=================================================================
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%\subsection{Frozen-core approximation}
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@ -380,18 +377,18 @@ with
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= \sum_{pqrs \in \BasFC} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
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\end{gather}
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\end{subequations}
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and the corresponding FC range-separation function \titou{$\rsmuFC{}{\Bas}(\br{}) = (\sqrt{\pi}/2) \WFC{}{\Bas}(\br{},\br{})$}.
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It is \titou{noteworthy} that, within the present definition, $\WFC{}{\Bas}(\br{1},\br{2})$ still tends to the regular Coulomb interaction \titou{as} $\Bas \to \infty$.
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and the corresponding FC range-separation function $\rsmuFC{}{\Bas}(\br{}) = (\sqrt{\pi}/2) \WFC{}{\Bas}(\br{},\br{})$.
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It is noteworthy that, within the present definition, $\WFC{}{\Bas}(\br{1},\br{2})$ still tends to the regular Coulomb interaction as $\Bas \to \infty$.
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Defining $\nFC{\modZ}{\Bas}$ as the FC (i.e.~valence-only) one-electron density obtained with a method $\modZ$ \titou{in $\Bas$}, the FC contribution of the complementary functional is then \titou{approximated by} $\bE{\LDA}{\Bas}[\nFC{\modZ}{\Bas},\rsmuFC{}{\Bas}]$ or $\bE{\PBE}{\Bas}[\nFC{\modZ}{\Bas},\rsmuFC{}{\Bas}]$.
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Defining $\nFC{\modZ}{\Bas}$ as the FC (i.e.~valence-only) one-electron density obtained with a method $\modZ$ in $\Bas$, the FC contribution of the complementary functional is then approximated by $\bE{\LDA}{\Bas}[\nFC{\modZ}{\Bas},\rsmuFC{}{\Bas}]$ or $\bE{\PBE}{\Bas}[\nFC{\modZ}{\Bas},\rsmuFC{}{\Bas}]$.
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%=================================================================
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%\subsection{Computational considerations}
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%=================================================================
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The most computationally intensive task of the present approach is the evaluation of $\W{}{\Bas}(\br{},\br{})$ at each quadrature grid point.
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Yet embarrassingly parallel, this step scales, in the general case (i.e.~$\wf{}{\Bas}$ is a \titou{multi}-determinant expansion), as $\Ng \Nb^4$ (where $\Nb$ is the number of basis functions in $\Bas$) but is reduced to $\order*{ \Ng \Ne^2 \Nb^2}$ when $\wf{}{\Bas}$ is a single Slater determinant.
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Yet embarrassingly parallel, this step scales, in the general case (i.e.~$\wf{}{\Bas}$ is a multi-determinant expansion), as $\Ng \Nb^4$ (where $\Nb$ is the number of basis functions in $\Bas$) but is reduced to $\order*{ \Ng \Ne^2 \Nb^2}$ when $\wf{}{\Bas}$ is a single Slater determinant.
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As shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, this choice for $\wf{}{\Bas}$ already provides, for weakly correlated systems, a quantitative representation of the incompleteness of $\Bas$.
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Hence, we will stick to this choice throughout the \titou{present} study.
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Hence, we will stick to this choice throughout the present study.
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In our current implementation, the computational bottleneck is the four-index transformation to get the two-electron integrals in the MO basis which appear in Eqs.~\eqref{eq:n2basis} and \eqref{eq:fbasis}.
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Nevertheless, this step usually has to be performed for most correlated WFT calculations.
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Modern integral decomposition techniques (such as density fitting \cite{Whi-JCP-73}) or atomic-orbital-based algorithms could be employed to significantly speed up this step.
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@ -461,24 +458,24 @@ iii) vanishes in the CBS limit, hence guaranteeing an unaltered CBS limit for a
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\label{fig:G2_Ec}}
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\end{figure*}
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We begin our investigation of the performance of the basis-set correction by computing the atomization energies of \ce{C2}, \ce{N2}, \ce{O2} and \ce{F2} obtained with Dunning's cc-pVXZ basis \trashPFL{sets} (X $=$ D, T, Q and 5).
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We begin our investigation of the performance of the basis-set correction by computing the atomization energies of \ce{C2}, \ce{N2}, \ce{O2} and \ce{F2} obtained with Dunning's cc-pVXZ basis (X $=$ D, T, Q and 5).
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\ce{N2}, \ce{O2} and \ce{F2} are weakly correlated systems and belong to the G2 set \cite{CurRagTruPop-JCP-91} (see below), whereas \ce{C2} already contains a non-negligible amount of strong correlation. \cite{BooCleThoAla-JCP-11}
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In a second time, we compute the atomization energies of the entire G2 set \cite{CurRagTruPop-JCP-91} composed by 55 molecules with the cc-pVXZ \titou{basis set family}.
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In a second time, we compute the atomization energies of the entire G2 set \cite{CurRagTruPop-JCP-91} composed by 55 molecules with the cc-pVXZ basis set family.
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This molecular set has been intensively studied in the last 20 years (see, for example, Refs.~\onlinecite{FelPetDix-JCP-08, Gro-JCP-09, FelPet-JCP-09, NemTowNee-JCP-10, FelPetHil-JCP-11, HauKlo-JCP-12, PetTouUmr-JCP-12, FelPet-JCP-13, KesSylKohTewMar-JCP-18}) and can be considered as a representative set of small organic and inorganic molecules.
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As a method $\modY$ we employ either CCSD(T) or exFCI.
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Here, exFCI stands for extrapolated FCI energies computed with the CIPSI algorithm. \cite{HurMalRan-JCP-73, GinSceCaf-CJC-13, GinSceCaf-JCP-15}
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We refer the interested reader to Refs.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJAc-JCTC-19} for more details.
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In the case of the CCSD(T) calculations, we have $\modZ = \ROHF$ as we use the restricted open-shell HF (ROHF) one-electron density to compute the complementary basis-set correction energy.
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In the case of exFCI, the one-electron density is computed from a very large CIPSI expansion containing several millions \trashPFL{of} determinants.
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In the case of exFCI, the one-electron density is computed from a very large CIPSI expansion containing several millions determinants.
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CCSD(T) energies are computed with Gaussian09 using standard threshold values, \cite{g09} while RS-DFT and exFCI calculations are performed with {\QP}. \cite{QP2}
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For the numerical quadratures, we employ the SG-2 grid. \cite{DasHer-JCC-17}
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Except for the carbon dimer where we have taken the experimental equilibrium bond length (\InAA{1.2425}), all geometries have been extracted from Ref.~\onlinecite{HauJanScu-JCP-09} and have been obtained at the B3LYP/6-31G(2df,p) level of theory.
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Frozen-core calculations are defined as such: a \ce{He} core is frozen from \ce{Li} to \ce{Ne}, while a \ce{Ne} core is frozen from \ce{Na} to \ce{Ar}.
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In the context of the basis-set correction, the set of active MOs, $\BasFC$, involved in the definition of the effective interaction \titou{[see Eq.~\eqref{eq:WFC}]} refers to the non-frozen MOs.
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||||
In the context of the basis-set correction, the set of active MOs, $\BasFC$, involved in the definition of the effective interaction [see Eq.~\eqref{eq:WFC}] refers to the non-frozen MOs.
|
||||
The FC density-based correction is used consistently when the FC approximation was applied in WFT methods.
|
||||
To estimate the CBS limit of each method, following Ref.~\onlinecite{HalHelJorKloKocOlsWil-CPL-98}, we perform a two-point X$^{-3}$ extrapolation of the correlation energies using the quadruple- and quintuple-$\zeta$ data that we add up to the HF energies obtained in the largest (i.e.~quintuple-$\zeta$) basis.
|
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As the exFCI \titou{atomization energies} are converged with a precision of about 0.1 {\kcal} \trashPFL{on atomization energies}, we can label \titou{these} as near-FCI.
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As the exFCI atomization energies are converged with a precision of about 0.1 {\kcal}, we can label these as near FCI.
|
||||
Hence, they will be our references for \ce{C2}, \ce{N2}, \ce{O2} and \ce{F2}.
|
||||
The results for these diatomic molecules are reported in Fig.~\ref{fig:diatomics}.
|
||||
The corresponding numerical data can be found in the {\SI}.
|
||||
|
||||
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